Theorem undif4 3608

Description: Distribute union over difference. (Contributed by NM, 17-May-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Assertion

Ref	Expression
undif4	⊢ ((A ∩ C) = ∅ → (A ∪ (B ∖ C)) = ((A ∪ B) ∖ C))

Proof of Theorem undif4

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		pm2.621 397	. . . . . . 7 ⊢ ((x ∈ A → ¬ x ∈ C) → ((x ∈ A ∨ ¬ x ∈ C) → ¬ x ∈ C))
2		olc 373	. . . . . . 7 ⊢ (¬ x ∈ C → (x ∈ A ∨ ¬ x ∈ C))
3	1, 2	impbid1 194	. . . . . 6 ⊢ ((x ∈ A → ¬ x ∈ C) → ((x ∈ A ∨ ¬ x ∈ C) ↔ ¬ x ∈ C))
4	3	anbi2d 684	. . . . 5 ⊢ ((x ∈ A → ¬ x ∈ C) → (((x ∈ A ∨ x ∈ B) ∧ (x ∈ A ∨ ¬ x ∈ C)) ↔ ((x ∈ A ∨ x ∈ B) ∧ ¬ x ∈ C)))
5		eldif 3222	. . . . . . 7 ⊢ (x ∈ (B ∖ C) ↔ (x ∈ B ∧ ¬ x ∈ C))
6	5	orbi2i 505	. . . . . 6 ⊢ ((x ∈ A ∨ x ∈ (B ∖ C)) ↔ (x ∈ A ∨ (x ∈ B ∧ ¬ x ∈ C)))
7		ordi 834	. . . . . 6 ⊢ ((x ∈ A ∨ (x ∈ B ∧ ¬ x ∈ C)) ↔ ((x ∈ A ∨ x ∈ B) ∧ (x ∈ A ∨ ¬ x ∈ C)))
8	6, 7	bitri 240	. . . . 5 ⊢ ((x ∈ A ∨ x ∈ (B ∖ C)) ↔ ((x ∈ A ∨ x ∈ B) ∧ (x ∈ A ∨ ¬ x ∈ C)))
9		elun 3221	. . . . . 6 ⊢ (x ∈ (A ∪ B) ↔ (x ∈ A ∨ x ∈ B))
10	9	anbi1i 676	. . . . 5 ⊢ ((x ∈ (A ∪ B) ∧ ¬ x ∈ C) ↔ ((x ∈ A ∨ x ∈ B) ∧ ¬ x ∈ C))
11	4, 8, 10	3bitr4g 279	. . . 4 ⊢ ((x ∈ A → ¬ x ∈ C) → ((x ∈ A ∨ x ∈ (B ∖ C)) ↔ (x ∈ (A ∪ B) ∧ ¬ x ∈ C)))
12		elun 3221	. . . 4 ⊢ (x ∈ (A ∪ (B ∖ C)) ↔ (x ∈ A ∨ x ∈ (B ∖ C)))
13		eldif 3222	. . . 4 ⊢ (x ∈ ((A ∪ B) ∖ C) ↔ (x ∈ (A ∪ B) ∧ ¬ x ∈ C))
14	11, 12, 13	3bitr4g 279	. . 3 ⊢ ((x ∈ A → ¬ x ∈ C) → (x ∈ (A ∪ (B ∖ C)) ↔ x ∈ ((A ∪ B) ∖ C)))
15	14	alimi 1559	. 2 ⊢ (∀x(x ∈ A → ¬ x ∈ C) → ∀x(x ∈ (A ∪ (B ∖ C)) ↔ x ∈ ((A ∪ B) ∖ C)))
16		disj1 3594	. 2 ⊢ ((A ∩ C) = ∅ ↔ ∀x(x ∈ A → ¬ x ∈ C))
17		dfcleq 2347	. 2 ⊢ ((A ∪ (B ∖ C)) = ((A ∪ B) ∖ C) ↔ ∀x(x ∈ (A ∪ (B ∖ C)) ↔ x ∈ ((A ∪ B) ∖ C)))
18	15, 16, 17	3imtr4i 257	1 ⊢ ((A ∩ C) = ∅ → (A ∪ (B ∖ C)) = ((A ∪ B) ∖ C))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∨ wo 357   ∧ wa 358  ∀wal 1540   = wceq 1642   ∈ wcel 1710   ∖ cdif 3207   ∪ cun 3208   ∩ cin 3209  ∅c0 3551

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-nul 3552

This theorem is referenced by: (None)